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Normalizer of maximal abelian subgroups of $$\mathrm{GL}(n)$$ and general hypergeometric functions. (English) Zbl 0848.33009
The generalized confluent hypergeometric systems on the Grassmannian have been formulated in the earlier papers of I. M. Gel’fand, V. S. Retakh, and V. V. Serganova [Generalized Airy functions, Schubert cells, and Jordan groups, Sov. Math., Dokl. 37, No. 1, 8–12 (1988); translation from Dokl. Akad. Nauk SSSR 298, No. 1, 17–21 (1988; Zbl 0699.33012)] and H. Kimura, Y. Haraoka, and K. Takano [Generalized confluent hypergeometric functions, Proc. Jap. Acad., Ser. A 68, 290–295 (1992; Zbl 0773.33004)]. The paper clarifies the symmetry of the confluent hypergeometric systems in terms of the action of the normalizer of maximal abelian subgroups of $$\mathrm{GL}(n)$$. The symmetry particularly reveals several transformation formulas for the classical confluent hypergeometric functions.
Reviewer: T.Sasaki (Kobe)

##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 33C70 Other hypergeometric functions and integrals in several variables 33C65 Appell, Horn and Lauricella functions